Abstract: In this talk we discuss the multidimensional utility maximisation problem with small nonlinear price impact. Using homogenisation techniques, we obtain a first order expansion of the utility function for small nonlinear price impact. We give a PDE characterisation of the utility loss for small nonlinear price impact and exhibit a family of asymptotically optimal strategies for the problem. This is a joint work in progress with Erhan Bayraktar and Thomas Caye.